atom

Let $P$ be a poset, partially ordered by $\le$. An element $a\in P$ is called an atom if it covers some minimal element of $P$. As a result, an atom is never minimal. A poset $P$ is called atomic if for every element $p\in P$ that is not minimal has an atom $a$ such that $a\le p$.

Examples.

1. Let $A$ be a set and $P=2^A$ its power set. $P$ is a poset ordered by $\subseteq$ with a unique minimal element $\mathrm{\varnothing }$. Thus, all singleton subsets $\{a\}$ of $A$ are atoms in $P$.

2. ${\mathbb{Z}}^{+}$ is partially ordered if we define $a\le b$ to mean that $a\mid b$. Then $1$ is a minimal element and any prime number $p$ is an atom.

Remark. Given a lattice $L$ with underlying poset $P$, an element $a\in L$ is called an atom (of $L$) if it is an atom in $P$. A lattice is a called an atomic lattice if its underlying poset is atomic. An atomistic lattice is an atomic lattice such that each element that is not minimal is a join of atoms. If $a$ is an atom in a semimodular lattice $L$, and if $a$ is not under $x$, then $a\vee x$ is an atom in any interval lattice $I$ where $x=\bigwedge I$.

Examples.

1. $P=2^A$, with the usual intersection and union as the lattice operations meet and join, is atomistic: every subset $B$ of $A$ is the union of all the singleton subsets of $B$.

2. ${\mathbb{Z}}^{+}$, partially ordered as above, with lattice binary operations defined by $a\wedge b=\gcd (a,b)$, and $a\vee b=\mathrm{lcm}(a,b)$, is a lattice that is atomic, as we have seen earlier. But it is not atomistic: $4$ is not a join of $2$’s; $36$ is not a join of $2$ and $3$ are just two counterexamples.

Title	atom
Canonical name	Atom
Date of creation	2013-03-22 15:20:09
Last modified on	2013-03-22 15:20:09
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	13
Author	CWoo (3771)
Entry type	Definition
Classification	msc 06A06
Classification	msc 06B99
Defines	atomic poset
Defines	atomic lattice
Defines	atomistic lattice
Defines	atomistic